Different versions of the nerve theorem and colourful simplices

Abstract

Given a simplicial complex and a collection of subcomplexes covering it, the nerve theorem, a fundamental tool in topological combinatorics, guarantees a certain connectivity of the simplicial complex when connectivity conditions on the intersection of the subcomplexes are satisfied.We show that it is possible to extend this theorem by replacing some of these connectivity conditions on the intersection of the subcomplexes by connectivity conditions on their union. While this is interesting for its own sake, we use this extension to generalize in various ways the Meshulam lemma, a powerful homological version of the Sperner lemma. We also prove a generalization of the Meshulam lemma that is somehow reminiscent of the polytopal generalization of the Sperner lemma by De Loera, Peterson, and Su. For this latter result, we use a different approach and we do not know whether there is a way to get it via a nerve theorem of some kind. versions, some of them with a connectivity condition in place of the acyclicity condition. It seems that the oldest reference to a nerve theorem with an acyclicity condition is due to Leray [9]. The case k = −1 is the nerve theorem for unions mentioned above.The second purpose of this paper is to provide a generalization of a related result -Meshulam's lemma [11, Proposition 1.6] and [10, Theorem 1.5] -which has several applications in combinatorics, such as the generalization of Edmonds' intersection theorem by Aharoni and Berger [1]. Meshulam's lemma is a Sperner-lemma type result, dealing with coloured simplicial complexes and colourful simplices, and in which the classical boundary condition of the Sperner lemma is replaced by an acyclicity condition. Its original proof relies on a certain version of the nerve theorem and we show that Theorem 1 can be used in the same vein to prove some variations of Meshulam's lemma. We also prove -with a completely different approach -the following generalization of this lemma, which can be seen as a homological counterpart of the polytopal Sperner lemma by De Loera, Peterson, and Su [5], in a same way Meshulam's lemma is a homological countepart of the classical Sperner lemma. It can also be seen as a homological counterpart of Musin's Sperner-type results for pseudomanifolds [15] and of Theorem 4.7 in the paper by Asada et al. [2]. We leave as an open question the existence of a proof based on a nerve theorem of some kind.We recall that a pseudomanifold is a simplicial complex that is pure, non-branching (each ridge is contained in exactly two facets), and strongly connected (the dual is connected). A colourful simplex in a simplicial complex whose vertices are partitioned into subsets V 0 , . . . , V m is a simplex with at most one vertex in each V i . Given a simplicial complex K and a subset U of its vertices, K[U] is the subcomplex induced by U, i.e. the simplicial complex whose simplices are exactly the simplices of K whose vertices are all in U.

Rainbow simplices in triangulations of manifolds

“…It is an important result from topological combinatorics with several applications in combinatorics, such as the generalization of Edmonds' intersection theorem by Aharoni and Berger [1] and many other results in which obtaining a system of distinct representatives is relevant, like for example, the Hall's theorem for hypergraphs [2]. Following this spirit, Meunier and Montejano [3] generalized Meshulam's lemma obtaining the following result which is the main tool in this paper to obtain rainbow simplices.…”

Electrocardiogram feature extraction and interval measurements using optimal representative cycles from persistent homology

“…Up to a scaling factor in the variable r , the geometric Čech complex of radius r is homotopy equivalent to the radius r Vietoris-Rips complex due to the Nerve Lemma [28]. Consequently, the definitions of the birth and death radii of an equivalence class of non-contractible loops presented in the “Example: a 1-dimensional topological feature in a simple dataset” section are equivalent to the definitions of the birth and death filtration of a class [ c ] ∈ H 1 ( V R r ( X )) given in Definition 0.4.…”